On the maximal \(p\)-ramified \(p\)-abelian extensions over \(\mathbb Z^d_p\)-extensions (Q1821147)

Let \(p\) be an odd prime, let \(k\) be a number field, and let \(k_\infty/k\) be a \(\mathbb Z^d_p\)-extension, so \(G=\text{Gal}(k_\infty/k)\simeq \mathbb Z^d_p\). Let \(\tilde X(k_\infty)\) denote the Galois group of the maximal abelian \(p\)-extension of \(k_\infty\) unramified outside the primes lying above \(p\). Let \(\Lambda_G\) denote the completed \(\mathbb Z^d_p\)-group ring of \(G\) and \(\rho(k_\infty)\) the rank of \(\tilde X(k_\infty)\) as a \(\Lambda_G\)-module. Let \(r_2(k)\) be the number of complex places of \(k\). The author considers the question of when \(\rho(k_\infty)=r_2(k)\). \textit{V. A. Babaitsev} [Math. USSR, Izv. 19, 1--12 (1982); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 45, 691--703 (1981; Zbl 0495.12012)] and \textit{R. Greenberg} have shown this is the case for almost all \(\mathbb Z_p\)-extensions of \(k\), and also when Leopoldt's conjecture holds for \(k\) [Invent. Math. 47, 85--99 (1978; Zbl 0403.12004)]. In the present paper, the author proves the following. Assume \(d=1\), \(k\) contains a primitive \(p\)-th root of unity, no prime of \(k\) above \(p\) splits completely in \(k_\infty/k\), and Iwasawa's invariant \(\mu(k_\infty/k)=0\). Then \(\rho(k_\infty)=r_2(k)\). The author also studies the behavior of \(\rho(k_\infty H_n)\) where \(H_n\) runs through the sequence of fields in a \(\mathbb Z_p\)-extension \(H_\infty/k\) disjoint from the \(\mathbb Z^d_p\)-extension \(k_\infty/k\), and shows that there exist non-negative integers \(r\) and \(c\) such that \(\rho(k_\infty H_n)=r p^n+c\) for all sufficiently large \(n\). Conditions implying \(c=0\) are given.